On the classification of solutions to a class of $N$-Liouville equations in ${\mathbb{R}}^N$

Given $N\ge 2$ and $\alpha >-1$, we consider the following weighted Liouville-type equation involving the $N$-Laplacian: \begin{equation*} \left{ \begin{aligned} -& Δ_N u = |x|^{Nα} e^u \quad $\text{ in }$ $\mathbb{R}$^N && , \ & \int_{$\mathbb{R}$^N} |x|^{Nα} e^u , dx < + \infty,. &&\end{aligned} \right. \end{equation*} Solutions have been completely classified when $N=2$ via complex analysis, and when $\alpha =0$ using Pohozaev identities and an isoperimetric argument. In this paper, we first devise a $P$-function approach to the classification result for all $\alpha >-1$ when $N=2$. Since it is not based on complex analysis, this alternative and more PDE-oriented approach naturally extends to $N\ge 3$ by providing the classification for any $-1<\alpha \le 0$. In particular, the explicit radial solutions are the unique ones for $-1<\alpha \le 0$ but become degenerate for special values ${\alpha }_k>0$, a hint that non-radial solutions might arise for $\alpha >0$ as it happens when $N=2$.